This page contains the following text (see the section “Higher-order busy beaver functions”):
At least, under any reasonable formulation of the notion of a higher-order Turing machine, well-orderings of ordinal type $\omega_1^{\text{CK}}$ are uncomputable even with respect to an $x$-th oracle Turing machine for any recursive ordinal $x$.
What about the other direction? That is, assuming that $\alpha$ is an arbitrarily large ordinal less than $\omega_1^{\text{CK}}$ and we know what it means for a Turing machine to have an oracle for a copy of some ordinal, is it necessarily true that for any copy $T$ of $\omega_1^{\text{CK}}$ the halting problem for the family of $\alpha$-th-order Turing machines is computable by some Turing machine that has access to an oracle for $T$?
I am inclined to think that the answer is “no, it is not true” (if the answer is “yes, it is true”, I would be surprised by how strong $\omega_1^{\text{CK}}$ is), but I cannot find the proof.